Let (An)
نویسندگان
چکیده
A central limit theorem for stochastic recursive sequences of topical operators. Abstract Let (A n) n∈N be a stationary sequence of topical (i.e. isotone and ad-ditively homogeneous) operators. Let x(n, x 0) be defined by x(0, x 0) = x 0 and x(n + 1, x 0) = A n x(n, x 0). It can modelize a wide range of systems including train or queuing networks, job-shop, timed digital circuits or parallel processing systems. When (A n) n∈N has the memory loss property, (x(n, x 0)) n∈N satisfies a strong law of large numbers. We show that it also satisfies the CLT if (A n) n∈N fulfills the same mixing and integrability assumptions that ensure the CLT for a sum of real variables in the results by An operator A : R d → R d is called additively homogeneous if it satisfies A(x + a1) = Ax + a1 for all x ∈ R d and a ∈ R, where 1 is the vector (1, · · · , 1) ′ in R d. It is called isotone if x ≤ y implies Ax ≤ Ay, in which the order is the product order on R d. It is called topical if it is isotone and homogeneous. The set of topical operators on R d will be denoted by T op d. We remind that the action of matrices with entries in the semiring R max = (R ∪ {−∞}, max, +) on R d max is defined by (Ax) i = max j (A ij + x j). When matrix A has no −∞-row, this formule defines a topical operator, also denoted by A. Such operators are called max-plus operators and operators composition corresponds to the product of matrices in the max-plus semiring. Let (A n) n∈N be a sequence of random topical operators on R d. A stochas-tic recursive sequence (SRS) driven by stochastic recursive sequence is a 1
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